Abstract
Modal analysis is widely used to decouple a set of linear second-order ordinary differential equations, identify mode shapes and natural frequencies and associate each mode shape with a particular natural frequency. However, this approach is not applicable to parametrically excited structural systems due to the presence of time-periodic coefficients. Furthermore, the nonlinear analysis of such structural systems using the theory of normal forms and invariant manifolds requires the linear part to be time-invariant. Motivated by these challenges, the present study develops dynamically equivalent, decoupled time-invariant forms of linear parametrically excited structural systems. First, equations of motion are temporarily written in the state-space form to facilitate the application of the Lyapunov-Floquet transformation. This transformation yields a time-invariant equation, which is then subjected to a decoupling transformation to obtain a new equation that can be expressed as a set of fully decoupled second-order time-invariant equations. The decoupling transformations are constructed based on the eigenvalues and their multiplicities of the time-invariant equation obtained from the Lyapunov-Floquet transformation. Decoupled time-invariant forms are generated for linear systems with and without external excitations. Examples are provided, and the results thus obtained are verified by numerical simulation.
